December 16, 2019

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Content of the Presentation

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  • Introduction
  • Methodology
  • Simulation Results
  • Discussion

Introduction

Multiple imputation

  • General statement:

\[\text{Missing data is ubiquitous.}\]

  • Ad hoc solutions may yield invalid inferences (Van Buuren 2018).

  • Rubin (1987) proposed the framework of MI.

Multiple imputation



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Algorithmic convergence

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Equations \(\widehat{R}\)

\[\begin{align} B&=\frac{T}{M-1} \sum_{m=1}^{M}\left(\bar{\theta}^{(\cdot m)}-\bar{\theta}^{(\cdot \cdot)}\right)^{2}, \quad \text{where} \quad \bar{\theta}^{(\cdot m)}=\frac{1}{T} \sum_{t=1}^{T} \theta^{(t m)}, \\ &\quad \text{ and } \quad \bar{\theta}^{(\cdot \cdot)}=\frac{1}{M} \sum_{m=1}^{M} \bar{\theta}^{(\cdot m)} \\ W&=\frac{1}{M} \sum_{m=1}^{M} s_{j}^{2}, \quad \text{where} \quad s_{m}^{2}=\frac{1}{T-1} \sum_{t=1}^{T}\left(\theta^{(t m)}-\bar{\theta}^{(\cdot m)}\right)^{2} \\ \widehat{R}&=\sqrt{\frac{\widehat{\operatorname{var}}^{+}(\theta | y)}{W}}, \quad \text{where} \quad \widehat{\operatorname{var}}^{+}(\theta | y)=\frac{N-1}{N} W+\frac{1}{N} B. \end{align}\]

Methodology

Mock code

Steps in the simulation study

simulation <- function() {
  mvtnorm(X,Z1,Z2) %>% 
  mutate(Y~X+Z1+Z2) %>% 
  for (max_iterations in 1:100) {
    ampute() %>% 
    impute() %T% 
    convergence_diagnostics %>% 
    lm(Y~X+Z1+Z2) %>% 
    pool %>% 
    simulation_diagnostics %>% 
    c(., convergence_diagnostics)
  }
}

replicate(simulation, n = 1000)

Simulation Results

Simulation diagnostics

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Convergence diagnostics

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Results

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Results

Simulation diagnostics versus convergence diagnostics

Inspect the autocorrelation dip

Discussion

Discussion

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Impression of ShinyMICE


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Inspect \(\widehat{R}\)

Inspect autocorrelation

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References

Rubin, Donald B. 1987. Multiple Imputation for Nonresponse in Surveys. Wiley Series in Probability and Mathematical Statistics Applied Probability and Statistics. New York, NY: Wiley.

Van Buuren, Stef. 2018. Flexible Imputation of Missing Data. Chapman; Hall/CRC.